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(* This is part of aac_tactics, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *)
(***************************************************************************)
(** * Tutorial for using the [aac_tactics] library.
Depending on your installation, either modify the following two
lines, or add them to your .coqrc files, replacing "." with the
path to the [aac_tactics] library. *)
Add Rec LoadPath "." as AAC_tactics.
Add ML Path ".".
Require Import AAC.
Require Instances.
(** ** Introductory example
Here is a first example with relative numbers ([Z]): one can
rewrite an universally quantified hypothesis modulo the
associativity and commutativity of [Zplus]. *)
Section introduction.
Import ZArith.
Import Instances.Z.
Variables a b c : Z.
Hypothesis H: forall x, x + Zopp x = 0.
Goal a + b + c + Zopp (c + a) = b.
aac_rewrite H.
aac_reflexivity.
Qed.
Goal a + c + Zopp (b + a + Zopp b) = c.
do 2 aac_rewrite H.
reflexivity.
Qed.
(** Notes:
- the tactic handles arbitrary function symbols like [Zopp] (as
long as they are proper morphisms w.r.t. the considered
equivalence relation);
- here, ring would have done the job.
*)
End introduction.
(** ** Usage
One can also work in an abstract context, with arbitrary
associative and commutative operators. (Note that one can declare
several operations of each kind.) *)
Section base.
Context {X} {R} {E: Equivalence R}
{plus}
{dot}
{zero}
{one}
{dot_A: @Associative X R dot }
{plus_A: @Associative X R plus }
{plus_C: @Commutative X R plus }
{dot_Proper :Proper (R ==> R ==> R) dot}
{plus_Proper :Proper (R ==> R ==> R) plus}
{Zero : Unit R plus zero}
{One : Unit R dot one}
.
Notation "x == y" := (R x y) (at level 70).
Notation "x * y" := (dot x y) (at level 40, left associativity).
Notation "1" := (one).
Notation "x + y" := (plus x y) (at level 50, left associativity).
Notation "0" := (zero).
(** In the very first example, [ring] would have solved the
goal. Here, since [dot] does not necessarily distribute over [plus],
it is not possible to rely on it. *)
Section reminder.
Hypothesis H : forall x, x * x == x.
Variables a b c : X.
Goal (a+b+c)*(c+a+b) == a+b+c.
aac_rewrite H.
aac_reflexivity.
Qed.
(** The tactic starts by normalising terms, so that trailing units
are always eliminated. *)
Goal ((a+b)+0+c)*((c+a)+b*1) == a+b+c.
aac_rewrite H.
aac_reflexivity.
Qed.
End reminder.
(** The tactic can deal with "proper" morphisms of arbitrary arity
(here [f] and [g], or [Zopp] earlier): it rewrites under such
morphisms ([g]), and, more importantly, it is able to reorder
terms modulo AC under these morphisms ([f]). *)
Section morphisms.
Variable f : X -> X -> X.
Hypothesis Hf : Proper (R ==> R ==> R) f.
Variable g : X -> X.
Hypothesis Hg : Proper (R ==> R) g.
Variable a b: X.
Hypothesis H : forall x y, x+f (b+y) x == y+x.
Goal g ((f (a+b) a) + a) == g (a+a).
aac_rewrite H.
reflexivity.
Qed.
End morphisms.
(** *** Selecting what and where to rewrite
There are sometimes several solutions to the matching problem. We
now show how to interact with the tactic to select the desired
one. *)
Section occurrence.
Variable f : X -> X.
Variable a : X.
Hypothesis Hf : Proper (R ==> R) f.
Hypothesis H : forall x, x + x == x.
Goal f(a+a)+f(a+a) == f a.
(** In case there are several possible solutions, one can print
the different solutions using the [aac_instances] tactic (in
proof-general, look at buffer *coq* ): *)
aac_instances H.
(** the default choice is the occurrence with the smallest
possible context (number 0), but one can choose the desired
one; *)
aac_rewrite H at 1.
(** now the goal is [f a + f a == f a], there is only one solution. *)
aac_rewrite H.
reflexivity.
Qed.
End occurrence.
Section subst.
Variables a b c d : X.
Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b.
Hypothesis H': forall x, x + x == x.
Goal a*c*d*c*d*b == a*c*d*b.
(** Here, there is only one possible occurrence, but several substitutions; *)
aac_instances H.
(** one can select them with the proper keyword. *)
aac_rewrite H subst 1.
aac_rewrite H'.
aac_reflexivity.
Qed.
End subst.
(** As expected, one can use both keywords together to select the
occurrence and the substitution. We also provide a keyword to
specify that the rewrite should be done in the right-hand side of
the equation. *)
Section both.
Variables a b c d : X.
Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b.
Hypothesis H': forall x, x + x == x.
Goal a*c*d*c*d*b*b == a*(c*d*b)*b.
aac_instances H.
aac_rewrite H at 1 subst 1.
aac_instances H.
aac_rewrite H.
aac_rewrite H'.
aac_rewrite H at 0 subst 1 in_right.
aac_reflexivity.
Qed.
End both.
(** *** Distinction between [aac_rewrite] and [aacu_rewrite]:
[aac_rewrite] rejects solutions in which variables are instantiated
by units, while the companion tactic, [aacu_rewrite] allows such
solutions. *)
Section dealing_with_units.
Variables a b c: X.
Hypothesis H: forall x, a*x*a == x.
Goal a*a == 1.
(** Here, [x] must be instantiated with [1], so that the [aac_*]
tactics give no solutions; *)
try aac_instances H.
(** while we get solutions with the [aacu_*] tactics. *)
aacu_instances H.
aacu_rewrite H.
reflexivity.
Qed.
(** We introduced this distinction because it allows us to rule
out dummy cases in common situations: *)
Hypothesis H': forall x y z, x*y + x*z == x*(y+z).
Goal a*b*c + a*c + a*b == a*(c+b*(1+c)).
(** 6 solutions without units, *)
aac_instances H'.
aac_rewrite H' at 0.
(** more than 52 with units. *)
aacu_instances H'.
Abort.
End dealing_with_units.
End base.
(** *** Declaring instances
To use one's own operations: it suffices to declare them as
instances of our classes. (Note that the following instances are
already declared in file [Instances.v].) *)
Section Peano.
Require Import Arith.
Instance aac_plus_Assoc : Associative eq plus := plus_assoc.
Instance aac_plus_Comm : Commutative eq plus := plus_comm.
Instance aac_one : Unit eq mult 1 :=
Build_Unit eq mult 1 mult_1_l mult_1_r.
Instance aac_zero_plus : Unit eq plus O :=
Build_Unit eq plus (O) plus_0_l plus_0_r.
(** Two (or more) operations may share the same units: in the
following example, [0] is understood as the unit of [max] as well as
the unit of [plus]. *)
Instance aac_max_Comm : Commutative eq Max.max := Max.max_comm.
Instance aac_max_Assoc : Associative eq Max.max := Max.max_assoc.
Instance aac_zero_max : Unit eq Max.max O :=
Build_Unit eq Max.max 0 Max.max_0_l Max.max_0_r.
Variable a b c : nat.
Goal Max.max (a + 0) 0 = a.
aac_reflexivity.
Qed.
(** Furthermore, several operators can be mixed: *)
Hypothesis H : forall x y z, Max.max (x + y) (x + z) = x+ Max.max y z.
Goal Max.max (a + b) (c + (a * 1)) = Max.max c b + a.
aac_instances H. aac_rewrite H. aac_reflexivity.
Qed.
Goal Max.max (a + b) (c + Max.max (a*1+0) 0) = a + Max.max b c.
aac_instances H. aac_rewrite H. aac_reflexivity.
Qed.
(** *** Working with inequations
To be able to use the tactics, the goal must be a relation [R]
applied to two arguments, and the rewritten hypothesis must end
with a relation [Q] applied to two arguments. These relations are
not necessarily equivalences, but they should be related
according to the occurrence where the rewrite takes place; we
leave this check to the underlying call to [setoid_rewrite]. *)
(** One can rewrite equations in the left member of inequations, *)
Goal (forall x, x + x = x) -> a + b + b + a <= a + b.
intro Hx.
aac_rewrite Hx.
reflexivity.
Qed.
(** or in the right member of inequations, using the [in_right] keyword *)
Goal (forall x, x + x = x) -> a + b <= a + b + b + a.
intro Hx.
aac_rewrite Hx in_right.
reflexivity.
Qed.
(** Similarly, one can rewrite inequations in inequations, *)
Goal (forall x, x + x <= x) -> a + b + b + a <= a + b.
intro Hx.
aac_rewrite Hx.
reflexivity.
Qed.
(** possibly in the right-hand side. *)
Goal (forall x, x <= x + x) -> a + b <= a + b + b + a.
intro Hx.
aac_rewrite <- Hx in_right.
reflexivity.
Qed.
(** [aac_reflexivity] deals with "trivial" inequations too *)
Goal Max.max (a + b) (c + a) <= Max.max (b + a) (c + 1*a).
aac_reflexivity.
Qed.
(** In the last three examples, there were no equivalence relation
involved in the goal. However, we actually had to guess the
equivalence relation with respect to which the operators
([plus,max,0]) were AC. In this case, it was Leibniz equality
[eq] so that it was automatically inferred; more generally, one
can specify which equivalence relation to use by declaring
instances of the [AAC_lift] type class: *)
Instance lift_le_eq : AAC_lift le eq := {}.
(** (This instance is automatically inferred because [eq] is always a
valid candidate, here for [le].) *)
End Peano.
(** *** Normalising goals
We also provide a tactic to normalise terms modulo AC. This
normalisation is the one we use internally. *)
Section AAC_normalise.
Import Instances.Z.
Require Import ZArith.
Open Scope Z_scope.
Variable a b c d : Z.
Goal a + (b + c*c*d) + a + 0 + d*1 = a.
aac_normalise.
Abort.
End AAC_normalise.
(** ** Examples from the web page *)
Section Examples.
Import Instances.Z.
Require Import ZArith.
Open Scope Z_scope.
(** *** Reverse triangle inequality *)
Lemma Zabs_triangle : forall x y, Zabs (x + y) <= Zabs x + Zabs y .
Proof Zabs_triangle.
Lemma Zplus_opp_r : forall x, x + -x = 0.
Proof Zplus_opp_r.
(** The following morphisms are required to perform the required rewrites *)
Instance Zminus_compat : Proper (Zge ==> Zle) Zopp.
Proof. intros x y. omega. Qed.
Instance Proper_Zplus : Proper (Zle ==> Zle ==> Zle) Zplus.
Proof. firstorder. Qed.
Goal forall a b, Zabs a - Zabs b <= Zabs (a - b).
intros. unfold Zminus.
aac_instances <- (Zminus_diag b).
aac_rewrite <- (Zminus_diag b) at 3.
unfold Zminus.
aac_rewrite Zabs_triangle.
aac_rewrite Zplus_opp_r.
aac_reflexivity.
Qed.
(** *** Pythagorean triples *)
Notation "x ^2" := (x*x) (at level 40).
Notation "2 ⋅ x" := (x+x) (at level 41).
Lemma Hbin1: forall x y, (x+y)^2 = x^2 + y^2 + 2⋅x*y. Proof. intros; ring. Qed.
Lemma Hbin2: forall x y, x^2 + y^2 = (x+y)^2 + -(2⋅x*y). Proof. intros; ring. Qed.
Lemma Hopp : forall x, x + -x = 0. Proof Zplus_opp_r.
Variables a b c : Z.
Hypothesis H : c^2 + 2⋅(a+1)*b = (a+1+b)^2.
Goal a^2 + b^2 + 2⋅a + 1 = c^2.
aacu_rewrite <- Hbin1.
rewrite Hbin2.
aac_rewrite <- H.
aac_rewrite Hopp.
aac_reflexivity.
Qed.
(** Note: after the [aac_rewrite <- H], one could use [ring] to close the proof.*)
End Examples.
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